Deriving what $\cos(A+2B)$ is

Question

We know that $\cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B)$ , but I don't understand with the $2B$. Would it just become $\cos(A+2B) = \cos(A)\cos(2B) - \sin(A)\sin(2B)$?

Thank you for any help you can give me.

"...would it just become...?" Yes, exactly. If desired, you can convert this into an expression involving trig functions around $B$ instead of $(2B)$, by simply further refining your expression via the corresponding formulas for $\cos(2B)$, and $\sin(2B).$ — user2661923, Jan 23 '21 at 20:19
There is nothing special about certain combinations of letters and numbers. $$\cos(\square+\circ) = \cos(\square)\cos(\circ) - \sin(\square)\sin(\circ)$$ — K.defaoite, Jan 26 '21 at 17:41

score 3 · Answer 1 · answered Jan 23 '21 at 20:20

Further expansion is possible with repeated use of the angle addition identities. $$\begin{align} \cos (a + 2b) &= \cos a \cos 2b - \sin a \sin 2b \\ &= \cos a (\cos b \cos b - \sin b \sin b) - \sin a (\sin b \cos b + \cos b \sin b) \\ &= \cos a (\cos^2 b - \sin^2 b) - \sin a (2 \sin b \cos b) \\ &= \cos a \cos^2 b - \cos a \sin^2 b - 2 \sin a \sin b \cos b. \end{align}$$

score 2 · Answer 2 · edited Jan 23 '21 at 20:35

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Yes, exactly! Just think of $2b=c$ and then substitute it back.

EDIT: It was marked that this not an answer, a requirement for substitution in general is that it has to be bijective, and indeed $2b=c$ is bijective.

Now you can use the formula $\cos(A+C) = \cos A\cos C - \sin A\sin C$ where $C=2B$ now you have it. Note that $\cos(2a)=\cos^2a-\sin^2(a)$. YoOu could use this as well later on.

edited Jan 23 '21 at 20:35

Bernard

175,478

answered Jan 23 '21 at 20:17

Maths

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This is by no way an answer! – Nicolas Jan 23 '21 at 20:19
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Whats wrong about it ? – Maths Jan 23 '21 at 20:20
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I think it's a perfectly fine answer I didn't even think about it that way ! – Tryingtolearnpython Jan 23 '21 at 20:21
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However, then you wouldn't be able to apply the double angle formulas which is a problem when it comes to simplification. – Tryingtolearnpython Jan 23 '21 at 20:21
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You do not bring any new information, only confirmed what was written in the original post. – Nicolas Jan 23 '21 at 20:22
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substitution was basically the answer – Maths Jan 23 '21 at 20:27
Your first, 10 words, answer was just confirming the substitution already made in the original post... – Nicolas Jan 23 '21 at 20:31

Deriving what $\cos(A+2B)$ is

2 Answers2